Conical calculus on schemes and perfectoid spaces via stratification
aa r X i v : . [ m a t h . A T ] A p r Conical calculus on schemes and perfectoid spacesvia stratification
Manuel Norman
Abstract
In this paper we show that, besides the usual calculus involving K¨ahlerdifferentials, it is also possible to define conical calculus on schemes andperfectoid spaces; this can be done via a stratification process. Followingsome ideas from [1-2], we consider some natural stratifications of thesespaces and then we build upon the work of Ayala, Francis, and Tanaka[3] (see also [4-5] and [18]); using their definitions of derivatives, smooth-ness and vector fields for stratified spaces, and thanks to some particularmethods, we are able to transport these concepts to schemes and perfec-toid spaces. This also allows us to define conical differential forms and theconical de Rham complex. At the end, we compare this approach withthe usual one, noting that it is a useful addition to K¨ahler method.
The concept of scheme was introduced by Grothendieck in his well known trea-tise EGA (see, for instance, [6]). A scheme is a locally ringed space which can becovered by affine schemes, that is, by locally ringed spaces which are isomorphicto the spectrum of some ring (the spectrum can be turned into a locally ringedspace using Zariski topology and a certain structure sheaf; see [7-8] for moredetails). More recently, Scholze defined in [9] the concept of perfectoid space,which is similar, from some points of view, to the notion of scheme. The ideais to assign to any perfectoid affinoid K -algebra ( R, R + ) a certain affinoid adicspace, namely Spa( R, R + ), which is called ’affinoid perfectoid space’; then, wedefine a perfectoid space to be an adic space over the perfectoid field K whichis locally isomorphic to some affinoid perfectoid spaces. For more details, werefer to [9-12] and [19]. These two notions are defined in such a way that they”locally resemble” some kind of space: schemes locally resemble affine schemes,while perfectoid spaces locally resemble affinoid perfectoid spaces. Another wellknown concept (that in fact can be define via ringed spaces, as schemes) is the Author:
Manuel Norman ; email: [email protected]
AMS Subject Classification (2010) : 57N80, 57R35, 14A15
Key Words : scheme, perfectoid space, calculus R n (which indeed is what a manifold locally looks like). Many gen-eralisations of manifolds arise in a similar way. Another idea of this type wasintroduced by tha author in [1] (and then developed in other papers): a struc-tured space locally resembles various kinds of algebraic structures. The theoryof structured spaces is not necessary to read this paper; however, we will followsome ideas from [1-2] (which are entirely reported here) and we will apply themin order to obtain a stratification of schemes and perfectoid spaces, that is, wewill show that there is a natural way to associate to these spaces a certain poset,which will then give us a poset-stratified space (see Definition 2.1.3 and Remark2.1.9 in [3]). Then, building upon [3], we will define derivatives over these strat-ifications, and this will allow us to extend the notion to schemes and perfectoidspaces (actually, the same method can be applied to any kind of space whichlocally resembles other spaces). We begin showing how we can stratify schemes and perfectoid spaces. Actually,a similar process can be applied to any notion of space which ”locally resembles”some other space. Let X be a scheme or perfectoid space, and consider someopen covering ( X p ) p by affine schemes or affinoid perfectoid spaces, that is, acollection of open affine schemes or affinoid perfectoid spaces such that S p X p = X . We define a map h : X → L as in Section 4 of [1], that is, we define: h ( x ) := { X t ∈ ( X p ) p : x ∈ X t } (2.1)Intuitively, this map measures ”how dense” is a point belonging to the underly-ing set of a scheme or perfectoid space w.r.t. the chosen cover (the dependenceon this cover may be removed in some cases; see Section 2.2). The collection L may be defined, as in [1], to be the ”power collection” of ( X p ) p without theempty sets, that is, the analogue of the power set, but with collection of sets,where we exclude the empty sets. Now the idea is to define, as in Section 4 of[1], the following preorder on X : x ≤ y ⇔ h ( x ) ⊆ h ( y ) (2.2)It is immediate to check that this is indeed a preorder, but it may not be apartial order. We define the abstract subset X/ ∼ of X , which clearly becomesa poset under the above ≤ , as the quotient of X by the following equivalencerelation: x ∼ y ⇔ h ( x ) = h ( y ) (2.3) In fact, the motivation of this paper is to show that also conical calculus (different fromthe ”usual” one with K¨ahler differentials) can be considered on schemes and perfectoid spaces. If not otherwise specified, when we say ’open covering’ we always refer to an open coveringw.r.t. the topology defined on the scheme or perfectoid space, and not to other topologiesthat will be defined later (this is why we will prefer to consider two topologies; see Remark2.1). X and thecorresponding poset X/ ∼ . More precisely, we recall (see, for instance, [13-16])that a (poset-)stratified space is a structure ( X, X s −→ P ), with X topologicalspace, P poset endowed with the Alexandroff topology, and s : X → P continu-ous surjection (there are some other slightly different notions, but here we willconsider this one). We can then define: Definition 2.1.
Let X be a scheme or a perfectoid space. Define the poset X/ ∼ via the equations (2.1) , (2.2) and (2.3) , and endow it with the Alexandrofftopology. A structure ( X, X s −→ X/ ∼ ) , where s : X → X/ ∼ is a continuoussurjection, and the underlying set X is also endowed with a second chosen topol-ogy (e.g. the smallest topology generated by the collection ( X p ) p ∪ { X, ∅} ), iscalled a ’stratification of the scheme/perfectoid space’. Remark 2.1.
The above definition needs some remarks. First of all, the small-est topology considered there is essentialy the same as the one in Example 1.1in [1] (this will allow us to follow an analoguous proof to the one in [2], whichwill give us an interesting example of stratification). Moreover, we can eitherdecide to drop the previous topology of X or to endow X with a second topol-ogy, turning it into a bitopological space (see [17]). In the latter case, we willalways refer (when dealing with X/ ∼ and with s ) to the second topology, e.g.the smallest one defined above, while we will always refer to the former whenconsidering the open coverings.Another important aspect to notice is that a stratification of a scheme (or per-fectoid space) is indeed a particular case of stratified space, as it is clear by itsdefinition.As announced in the above Remark, we now give an explicit example ofstratification, which will be regarded as the ’standard one’. The proof of thisexample is almost the same as the one of Proposition 5.1 in [2]; we rewrite ithere. Example 2.1.
Let X be any scheme of perfectoid space, and consider somecovering ( X p ) p . Define the map s : X → X/ ∼ by s ( x ) := [ x ], and suppose thatthe covering is such that sup x ∈ X | h ( x ) | < ∞ . If we let the second topology bethe smallest one generated by the covering (see the example in Definition 2.1),then ( X, X s −→ X/ ∼ ) is a stratification of the scheme (or perfectoid space). Toprove this, we need to show that s is continuous and surjective. Surjectivityis clear, since each [ x ] ∈ X/ ∼ is reached at least by x . To prove continuity,first notice that, by definition of Alexandroff topology, whenever U is an opensubset of X/ ∼ and whenever t ∈ U , r ∈ X/ ∼ , we have r ∈ U . Now, consider s − ([ x ]). This is equal to the set: s − ([ x ]) = { y ∈ X : h ( y ) = h ( x ) } Here, we define the cardinality in the same way as for sets. For instance, let A , B and C be three different (i.e. they are not equal to each other) sets, and let C be the collectioncontaining these sets. Then, |C| = 3. h ( x ) = { X t } some X t ’s in ( X p ) p then we know that all the x ∈ s − ([ x ]) belong to T X t X t , with the same X t ’s asbefore. Actually, we can even be more precise: we can delete the other X i ’s in( X p ) p that intersect T X t X t but do not belong to the collection of X t ’s above.This means that: s − ([ x ]) = ( \ X t X t ) \ ( [ X i not belongingto the previouscollection of X t ’s X i )If U is an open subset of X/ ∼ , by what we said above we have that, whenever x ∈ U , all the points y ∈ X/ ∼ such that x ≤ y belong to U . This implies,by definition of ≤ , that the set s − ( U ) is the union of some intersections ofsets in ( X p ) p : indeed, the gaps due to the differences of sets are filled because h ( x ) ⊆ h ( y ) and by the previous discussion. Consequently, s − ( U ) can bewritten as a union of some intersections as the above one. But by assumptionwe have intersections of a finite number of elements X t , which are open bydefinition of the (second) topology on X . Thus, these intersections are open,and the union of these open sets is open. This proves the continuity of s , andthus we have verified that ( X, X s −→ X/ ∼ ) is a stratification of the scheme (orperfectoid space) X .Clearly, stratifications can be used to study schemes and perfectoid spacesfrom other perspectives. We only sketch some of the possible results, eventhough they are not needed in the rest of this paper. Section 2.2 containsinstead an important and useful ”refinement”: using direct limits (wheneverpossible), we will avoid the dependence on the covering ( X p ) p . Consider any collection of objects taken from the category of schemes (or fromthe category of perfectoid spaces over some fixed K ), and assign to each elementof the family one and only one stratification ( X, X s −→ X/ ∼ ) (we maintainthem fixed throughout the discussion). Then, consider the class of all thesestratifications; they will be the objects in a new category, where the morphismsare all the usual ones between stratified spaces (not anymore the ones betweenschemes or perfectoid spaces). Notice that there is a bijection between thechosen collection and the new category, because each stratification also involvesthe scheme or perfectoid space itself.This category is actually a full subcategory of Strat (see Section 4.2 in 14]), asit can be easily verified. Consequently, some results for
Strat still hold also onthis subcategory. In particular, it can be seen that the following result holds:
Proposition 2.1.
Every subcategory of
Strat constructed as above and equippedwith the class of weak equivalences is a homotopical category.
Proof.
See Lemma 4.3.7 in [14].Thus, it is possible to construct a homotopical category starting from anycollection of schemes or perfectoid spaces over the same K , in such a way thatthere is a bijection between its objects and the elements in the chosen family.Another possible kind of stratification can be obtained slightly changing Def-inition 2.1. First of all, here we will endow in some way (below, two possiblemethods are shown) the space X with a partial order. Then (after endowingit with the Alexadroff topology), we will consider the poset X instead of X/ ∼ (this is the slight generalisation of Definition 2.1, which leads to other possiblekinds of stratifications of schemes and perfectoid spaces; clearly, such a struc-ture is still a stratified space). More precisely, we know that X is a preorderedset under ≤ . It is possible to define a partial order (cid:22) on X in the followingstandard way: x ≺ y ⇔ x ≤ y and not y ≤ xx (cid:22) y ⇔ x ≺ y or x = y Another possible method to obtain a preorder is to start with some stratification(
X, X s −→ X/ ∼ ) and to define the following preorder (see Construction 4.2.3 in[14]): x ≤ s y ⇔ sx ≤ sy which can be turned into a partial order as in the previous situation. In anycase, we obtain some poset structure for X . Now consider the stratified space( X, X i −→ X ) (where i denotes the identity map). Then we have the followingresult: Proposition 2.2.
Let X be a scheme or a perfectoid space, and consider thestratified space ( X, X i −→ X ) , where X is endowed with a poset structure eithervia the partial order obtained from ≤ or via the partial order obtained from ≤ s (for some other stratification of X related to s ). Then, the following three re-sults hold:(i) ( X, X i −→ X ) is a fibrant stratified space;(ii) the stratified geometric realisation of the nerve of ( X, X i −→ X ) is a fibrantstratified space;(ii) ( X, X i −→ X ) and the stratified geometric realisation of its nerve are homo-topically stratified spaces.Proof. (i) is Example 4.3.4 in [14], while (ii) is Example 4.3.5 in the same paper.(iii) is obtained applying Theorem 4.3.29 in [14] to the previous two results.It is also possible to associate to schemes and perfectoid spaces other kindsof spaces, as shown below. Proposition 2.3.
Let X be a scheme or a perfectoid space and consider astratification ( X, X s −→ X/ ∼ ) (or even the more general kind of stratification ( X, X i −→ X ) ). Then, we can define the following spaces: (i) the simplicial set SS ( X ) , whose n -simplices are given by Strat ( k ∆ n k , X ) ;(ii) the prestream ( X, ≤ | • ) (or ( X, ≤ s | • );(iii) the d-space ( X, d ≤| • X ) (or ( X, d ≤ s | • X ) ).Proof. For (i), see Definition 7.1.0.3 in [16]. For (ii), see 5.1.7 in [14] (thisprestream is simply obtained by restriction on each open subset U of X ). For(iii), see 5.1.11 in [14].This allows us to study schemes and perfectoid spaces also from the pointsof view of simplicial sets, streams and d-spaces. A stream is usually defined tobe a particular kind of prestream; see, for instance, Definition 5.1.14 in [14] forHaucourt streams and Remark 5.1.19 in the same paper for Krishnan streams.We will not go deeper into these topics here. The results in this subsection can also be applied to Section 2.1. The ideais to define a direct limit in order to avoid the dependence on ( X p ) p in theconstruction of a stratification. Given a scheme or a perfectoid space X , considersome open covering ( X p ) p . A refinement of such a cover is another open cover( Y t ) t of X such that: ( X p ) p ⊆ ( Y t ) t This means that ( Y t ) t contains all the affine schemes in the covering ( X p ) p ,with some possible additional affine schemes. Of course, the same argumentholds for perfectoid spaces and affinoid perfectoid spaces. The reason why wedo this is due to h : more affine schemes in the covering can give, in general,more interesting posets X/ ∼ . Remark 2.2.
As we had already noticed, we remark again the the term ’open’refers here to the topology defined on the scheme or perfectoid space X , andnot to other topologies (that is, not to the second topology, which actually atthis time has not been defined yet).Now consider for each open covering, say ( X rp ) p , the corresponding poset X/ ∼ r . Since posets form a category, we can define a direct limit as follows. Itis clear that the following implication holds:( X rp ) p ⊆ ( X tp ) p ⇒ X/ ∼ r ⊆ X/ ∼ t (where we consider the same representatives of X/ ∼ r also on X/ ∼ t , exceptwhen not possible, of course). Thus, the set of all the open covers (by affineschemes or affinoid perfectoid spaces) for X is an index set and the family of allthe corresponding posets is indexed by it. We consider the inclusion morphisms: ι : X/ ∼ r → X/ ∼ t for ( X rp ) p ⊆ ( X tp ) p . It is clear that all the necessary conditions are satisfied, andwe can thus consider (when it exists) the following direct limit, which avoids6anuel Norman Conical calculus on schemes and perf. spaces via stratif.the dependence on the chosen cover:lim −→ ( Xrp ) p X/ ∼ r If this limit exists, it will be called the ’refined corresponding poset of thescheme/perfectoid space’, and we will usually consider it instead of any othercovering.
Remark 2.3.
Direct limits involving open covers are also used, for instance,when defining the ”refined” ˇCech cohomology, that is, ˇCech cohomology thatdoes not depend on the chosen cover. However, in our case we consider adifferent approach: for reasons due to the definition of h and hence of X/ ∼ ,here it is more interesting to consider a refinement to be a covering with moreelements than the previous one, and with at least all the previous affine schemesor affinoid perfectoid spaces. Instead, with ˇCech cohomology we consider arefinement to be a cover whose elements are subsets of some other elements inthe other cover (see, for instance, Chapter 10 in [22]). The limit above may notexist, and in such cases we will unfortunately have a dependence on the chosencovering. However, we will see later that the definition of derivative actuallydepends on the considered stratification (hence, also on the chosen map s , notonly on the poset X/ ∼ ): this can be seen similarly to the dependence on thechosen direction for directional derivatives. Thus, the choice of the covering willnot cause problems, since the ”stratified derivative” will always depend on asort of ”direction” (in this case, the stratification).We also note that this definition of refinement does not lead, in general, to a”degenerate” poset, that is, a poset which turns out to actually be X itself:indeed, since the notion of ’open’ depends on the chosen topology, this couldonly happen with a discrete topology. We will use the degenerate case in thedefinition of derivative, because in such situations it turns out to be reallyuseful: it allows us to obtain a map between schemes or perfectoid spaces froma stratified map.Now that we have prepared the groud for the application of the work ofAyala, Francis and Tanaka [3], we briefly review the fundamental part of theirpaper which allows us to finally define derivatives on schemes and perfectoidspaces (and actually, as previously remarked, also on any kind of space to whichthe arguments of this section can be applied). We start this section recalling the notion of derivative for stratified spaces de-fined in [3]. We will then define a map that assigns to each f : X → X (mapsbetween schemes or perfectoid spaces) a function from the chosen stratificationof X and the chosen stratification of X (again denoted by f ), we will then7anuel Norman Conical calculus on schemes and perf. spaces via stratif.extend this function and derive it, and we will finally define the derivative of f : X → X using the above map.Following Section 3.1 in [3], consider some compact stratified space X , andconsider the stratified space R i × C ( X ), where the cone C ( X ) → C ( P ) is definedas in Definition 2.1.14 in [3]: C ( X ) := ∗ a { }× X R ≥ × X (3.1)and C ( P ) := ∗ a { }× P [1] × P (3.2)The space R i × C ( X ), which will be indicated by U , is composed by points thatwill be denoted by ( x, [ y, z ]), with ( x, y, z ) ∈ R i × R ≥ × X . Thanks to thefollowing identification, where T M denotes the tangent bundle of the manifold M : T R i × C ( X ) ∼ = R iv × R i × C ( X ) = R iv × U (where the points are indicated by ( v, x, [ y, z ])), we have a homeomorphism γ : R > × T R i × C ( X ) → R > × T R i × C ( X ) given by:( a, v, x, [ y, z ]) γ ( a, av + x, x, [ ay, z ]) γ can also be seen as a map γ a,x , as explained at the beginning of Section 3.1in [3].Before proceeding, we need to define: Definition 3.1.
A continuous stratified map f between two stratified spaces ( X, X s −→ P ) , ( Y, Y s −→ P ) is a commutative diagram of this kind: X → Y ↓ ↓ P → P Now consider a continuous stratified map f between two compact stratifiedspaces such that C ( P ) → C ( P ) sends the cone point to the cone point. Therestriction to the cone point stratum is denoted by f | R i . The map f ∆ (seeDefinition 3.1.2 in [3]) is then given by: f | ∆ := id R > × f | R i × f Notice that this is not so restrictive when considering our particular case of stratificationsof schemes and perfectoid spaces. Indeed, by Definition 2.1 we know that we can choose anypossible second topology on X , so we just need to consider one for which X is compact. Forexample, the topology in Example 1.1 in [1] can be often used, because in many cases X turnsout to be compact. If X = ∅ , [ y, z ] = ∗ . Definition 3.2 (Derivative of stratified maps) . Let ( X, X s −→ P ) , ( Y, Y s −→ P ) be two compact stratified spaces and let f be a continuous stratified mapbetween R i × C ( X ) and R j × C ( Y ) . f is continuously derivable along R i (or,equivalently, f is C along R i ), if C ( P ) → C ( P ) sends the cone point to thecone point and if there is a continuous extension (which, if it exists, is unique) e Df : R ≥ × T R i × C ( X ) e Df R ≥ × T R j × C ( Y ) ↑ ↑ R > × T R i × C ( X ) γ − ◦ f ∆ ◦ γ R > × T R j × C ( Y ) The restriction to a = 0 is denoted by Df . D x f is defined as the compositionof the projection map onto the second term and the map from R iv × { x } × C ( X ) to R jw × { f ( x, ∗ ) } × C ( Y ) . For n > , a map is continuously derivable along R i n times (or, equivalently, it is C n along R i ) if it is continuously derivablealong R i and if Df is continuously derivable n − times along R i × R i . If f is continuously derivable n times along R i , ∀ n , then f is C ∞ along R i (or,equivalently, it is conically smooth along R i ). We can now finally apply the previous definitions to our case. We firstneed to assign to each map between schemes or perfectoid spaces a certain mapbetween their chosen stratifications. Actually, we will do this in two steps: wefirst assign, in a uniquely determined way, a continuous stratified map to ourmap of schemes or perfectoid spaces; then, we assign in a certain way anothercontinuous stratified map, we derive it and we transform it into a map betweenschemes or perfectoid spaces. We then conclude noting a useful generalisation,which can be regarded as the actual way to define the derivative. Start witha map f : X → X between schemes or perfectoid spaces. By definition, wecan take the map between the underlying sets, denoted again by f . Stratify insome way the two spaces, say with s , s , respectively. Then we can write thefollowing diagram, where we should find a map g for which it is commutative,i.e. g ◦ s = s ◦ f : X f −→ X ↓ s ↓ s X / ∼ g −→ X / ∼ We have already fixed, as usual, the representatives of the equivalence classesof X i / ∼ i . We would like to do the following: g ( s ( x )) = s ( f ( x )) g ( y ) = s ( f ( s − ( y )))9anuel Norman Conical calculus on schemes and perf. spaces via stratif.The unique problem is that s − ( y ) is a set. Of course, if f is constant on eachof these sets, then everything works properly, but this is really restrictive, sowe will need to do something more. The idea is to choose a representative foreach s − ( y ), and to consider a new diagram. More precisely, choose one andonly one representative for each s − ( y ) (notice that these sets form a partitionof X ), which will be denoted by t y . Now let R s X denote the subspace of all thechosen representatives t y , for y ∈ X / ∼ ( R stays for ’representatives’). Wecan endow this space with the subspace topology, so that the restrictions of themaps will still be continuous. We have thus arrived at the following diagram: R s X f −→ X ↓ s ↓ s X / ∼ g −→ X / ∼ where f and s are actually restricted to their new domain. Now, we can define g uniquely (up to the choice of the representatives) as follows: g := s ◦ f ◦ s − (3.3)which clearly assures the commutativity of the new diagram. Notice that thiscan be certainly done because s is a bijection between X / ∼ and R s X . There-fore, this function is a continuous stratified map associated to f . By Definition3.2, we need to ”extend” this map to another domain. We will do this in thefollowing way:1) We consider i = j ;2) The extension of the continuous stratified map obtained before is the ”obvi-ous one”, that is, the component in the cone of R s X (respectively, X ) is sent tothe component of the cone of X / ∼ (respectively, X / ∼ ) as described in [3](recall that the cone is a stratified space); the first component is simply w ; thefunction on the second component (which comes from (3.1)), that is, the firstcomponent of the cone, is simply a (with a ≥ f issimilar to the first ones above, i.e. the first two components are as above, whilethe third component is the obvious one .This gives us another continuous stratified map, say b f , as requested by Defi-nition 3.2. Indeed, it is not difficult to see that the diagram obtained is stillcommutative. We then consider, if it exists, the derivative e D b f (or its restrictionto a = 0, denoted by D b f ). Remark 3.1.
Since the definition of derivative also involves R ≥ , R > and R i ,we will also have some functions on the corresponding components, as notedabove. These can be, for instance, simply w and a , as in the previous definition. For this last extension, recall the notion of ’cone functor’, which is the functor C : Top → Top given by C f : C ( X ) → C ( Y ), which is defined for continuous maps f : X → Y by C f ([ x, y ]) := [ f ( x ) , y ]. { x } , which is open, is an affinescheme. This will imply the statement above. We will prove that each { x } isisomorphic, as a locally ringed space, to Spec( R ). Since R is not only a ring,but also a field, Spec( R ) is a singleton (its only prime ideal is { } , which is infact its only element), and moreover we have: O Spec( R ) ( ∅ ) = { }O Spec( R ) ( { } ) = R { x } is a ring (whose additive and multiplicative identity coincide, and are = x )which is clearly isomorphic to Spec( R ), since they are both singletons. It remainsto find isomorphisms: φ U : O Spec( R ) ( U ) → O { x } ( ρ − ( U ))for each open U in Spec( R ), and where ρ ( x ) = { } . When U = ∅ , we clearly havethe zero isomorphism between the two singleton rings. When U = { } , we cantake the isomorphism sending each r ∈ R to itself. The commutativity propertyrequired by the definition of morphism of locally ringed spaces is obviouslysatisfied, because of the trivial morphisms involved. Thus, { x } is isomorphic toSpec( R ) and the domain and codomain of the derivative can be seen as schemes.We define the n -th derivative of f as the ( n − Df (viewed as amap between schemes/perfectoid spaces). A map that can be derived infinitelymany times is called conically smooth along R i . Remark 3.2.
It is important to note that the n -th derivative of the stratifiedmap obtained from some f as above is, in general, different from the ( n − Df viewed as a map between stratified spaces.We conclude this section with some examples. Example 3.1.
Consider some scheme or perfectoid space X , and let X/ ∼ beits corresponding poset. Stratify this space via the standard s (see Example2.1; here we assume that we are dealing with some covering for which s is11anuel Norman Conical calculus on schemes and perf. spaces via stratif.continuous, as in such example), choose as representatives t y for the sets s − ( y )precisely the same as the representatives in X/ ∼ , so that R sX is the same set as X/ ∼ , even though these spaces are endowed with different topologies ( R sX hasthe subspace topology, while X/ ∼ has Alexandroff topology). Consider anymap f : X → R sX (where the codomain is endowed with the trivial topology,we consider the trivial covering via all its singletons and we stratify it via theidentity) defined as follows: f ( t y ) := t y while f can be defined in any way for the other points in X . Then, in order toobtain a continuous stratified map, we need to find a function g for which thefollowing diagram commutes: R sX f −→ R sX ↓ s ↓ s = iX/ ∼ g −→ X/ ∼ Indeed, the set R sX / ∼ is the same as the set X/ ∼ . This is because thedefinition of ∼ together with the trivial covering of R sX imply that R sX / ∼ isequal to R sX , which is thus equal (as a set) to X/ ∼ , as we noted above.Via (3.3), we conclude that g is the identity map. This statement follows fromthe fact that the restriction of s to R sX is clearly the identity, and the sameholds true for f . Thus, we have: R sX i −→ R sX ↓ i ↓ iX/ ∼ g −→ X/ ∼ from which we clearly have g = i , where i denotes the identity. Consequently,extending the identity map in the obvious way (as previously outlined), weobtain again the identity, whose derivative (by Example 3.1.7 in [3]) is the iden-tity map (which can be easily turned into a map between schemes or perfectoidspaces). Example 3.2.
For the meaning of ’extension’ in this example, see also Remark3.1. Making use again of Example 3.1.7 in [3], any map between schemes orperfectoid spaces which is extended, when defining the derivative, to somethingof the form b f ( x, y, z ) = ( k ( x ) , y, ρ y ( z ))has the following expression for the derivative at a = 0: D b f (1 , v, x, [ y, z ]) = (1 , Dk x ( v ) , k ( x ) , [ x, ρ ( z )])where v is the same as v in the identification after equation (3.2). As usual, thismap can be turned into a map between schemes or perfectoid spaces.12anuel Norman Conical calculus on schemes and perf. spaces via stratif. Remark 3.3.
We remark that here we have considered only one possible kindof conical derivative. It is clear that, if we find other ways to assign a continuousstratified map to our f , then we can define the derivative analogously to whatwe did above. It seems that our map is quite natural, even though we often needto restrict the domain. It would be interesting to find some conditions whichassure that g actually exists without using such restriction, and then definethe derivative in those cases (notice that, for instance, with the identity therewould be no need to restrict the domain: g = i would work properly, yeldingas derivative the identity map, which in fact is conically smooth along R i ).Another natural kind of derivative is, for example, the following one. Considerthe stratification ( X , X i −→ X ), with the partial order obtained from thespecialisation preorder on X ; we have the diagram: X f −→ X ↓ i ↓ s X g −→ X / ∼ which is certainly commutative if we define g := s ◦ f without using any restriction. Then we can proceed as above and evaluate Df .Notice that we do not have a dependence on the first stratification here, becauseit has been fixed.Furthermore, there is also another way (which in some cases is actually betterthan the discrete one) to endow the derivatives with the structure of mapsbetween schemes or perfectoid spaces. If we view the space X as a subset ofthe domain and codomain of the derivative via isomorphism, then X is disjointfrom the difference of these spaces and X itself, and we can use the extensiontopology (see [31] or Section 4 in [1]) to endow them with a natural extension ofthe first and second topologies on X . The problem is then that the new spaceobtained should be a scheme or a perfectoid space, and this is not guaranteed. We now discuss conical vector fields and conical differential forms. The defini-tions needed for this section are quite involved, so to refer to [3] (and actuallyalso to [4-5] and [18]) instead of writing them here. A stratification of a schemeor perfectoid space is called C if the stratified space is C . It may be helpfulto redefine the second topology on X so that it becomes paracompact, in caseit were not. The definition of conically smooth stratification is analogous. Aconical vector field on some conically smooth stratification of a scheme or per-fectoid space X is an element of the vector space Θ( X ) of parallel vector fields13anuel Norman Conical calculus on schemes and perf. spaces via stratif.defined in Definition 8.1.2 of [3]. Flows of conical vector fields are defined belowit. The idea of the definition of conical differential form is the same as the usualone (see, for instance, [23]). Since Θ( X ) is a vector space, we can consider itsdual, that is, the space Θ ∗ ( X ) consisting of conical covector fields. As covectorfields are also called differential forms, conical covector fields are called conicaldifferential forms. We define conical k -forms on X as the elements of the space:( k ^ Θ( X )) ∗ A notion of wedge product for conical forms can be defined as usual. We wouldalso like to define a sort of ’exterior derivative’. For conical 1-forms, this is nota problem: by Section 20 in [23], we can define it via a formula that holds forthe usual differential 1-forms dω ( X , X ) := X ω ( X ) − X ω ( X ) − ω ([ X , X ]) (4.1)where the Lie bracket can be defined in the usual way. For n >
1, we recall thefollowing formula for the Lie derivative of differential forms, which will be usedbelow: L X ω ( X , ..., X n ) = X ( ω ( X , ..., X n )) − n X i =1 ω ( X , ..., [ X, X i ] , ..., X n )Moreover, we have Cartan formula: L X ω = i X ( dω ) + d ( i X ω )If we define the interior product as usual, we can thus define the exterior deriva-tive of conical n -forms, for n >
1, as follows: d ( i X ( ω ( X , ..., X n ))) := X ( ω ( X , ..., X n )) − n X i =1 ω ( X , ..., [ X, X i ] , ..., X n ) − i X ( d ( ω ( X , ..., X n ))) (4.2)With X variable in the space of vector fields, we obtain the desired definitioninductively (recall that n = 1 has already been defined). Some natural questionsarise, and these stimulate further research: Question 4.1.
Is this the ”best” definition possible?This question may actually depend also on the following one:
Question 4.2.
Is it possible to find local expressions of differential forms, asfor manifolds?Relatively to Question 4.2, we notice that it could be possible to locallydefine the exterior derivative via such local expressions. Moreover, a possible14anuel Norman Conical calculus on schemes and perf. spaces via stratif.way to answer this question could be found considering the strata of parallelvector fields and applying the theory developed in [3-5] and [18].We conclude this section noting that the above definition of exterior derivativeallows us to define a new de Rham complex, which will be called ’conical deRham complex’:
Theorem 4.1 (Conical de Rham complex) . Analogously to the classical case,we can define a ’conical de Rham complex’ via the above definition of exteriorderivative. More precisely, we have: → Θ( X ) ∗ d −→ Θ ( X ) ∗ d −→ Θ ( X ) ∗ d −→ ... (4.3) where the spaces involved are the vector spaces Θ n ( X ) ∗ := ( n ^ Θ( X )) ∗ and where the cohomology groups are given by the quotients of closed forms byexact forms (as usual).Proof. The fact that d ◦ d = 0 can be easily proved starting from the definitionabove (it is only a long calculation). By this fact, it clearly follows that exactforms are closed (where these notions are defined similarly to the classical case).Then, we clearly have a complex which is analogous to de Rham complex, butinvolving conical forms.An important question naturally arises: Question 4.3.
Is there any connection between conical de Rham complex andK¨ahler-de Rham complex?This could also give an answer to Question 4.1.
We now briefly compare the usual method with K¨ahler differentials and theconical approach. It is clear that K¨ahler approach is a bit involved: it usesvarious concepts, while the definition of derivative along R i is easier to state.Furthermore, the module of R -derivations is in general more difficult to com-pute (and this is one of the reasons why we consider K¨ahler differential forms,see also [20]), while the extension in Definition 3.2 can be simpler, since it isonly one kind of derivative of a function, and not the module of all the possiblederivations . In fact, the introduction of derivations was due to the problem There are some exceptions: for instance, when we consider polynomials (see [20] for someexamples) we usually have simple calculations, as in classical calculus. Of course, the difficultyarises in more general cases. In such situations, the conical approach can be easier. Even for higher order derivations (see, for instance, [29-30]) we have some structures,usually groups or algebras, which may be difficult to compute. Again, in these situations theconical approach is often easier, if we only want to evaluate derivatives.
In this paper we have shown a conical approach to calculus on schemes andperfectoid spaces. Actually, the stratification method can also lead to otherinteresting results, not necessarily related to calculus (for some examples, seeSection 2.1). Building upon [3], we have defined a notion of derivative which issimpler than the one involving the computation of a module, or a group/algebrain the higher order case, of all the possible derivations. Moreover, when we con-sider differential forms, both K¨ahler and conical approach give rise to interestingresults, notably two kinds of de Rham complexes. Conical calculus thus turnsout to be a really interesting and useful addition to the usual K¨ahler method.Some directions for future works are given in Section 4.